A257602 Expansion of (1 + x + 21*x^2 + x^3 + x^4)/(1 - x)^5.

1, 6, 41, 156, 426, 951, 1856, 3291, 5431, 8476, 12651, 18206, 25416, 34581, 46026, 60101, 77181, 97666, 121981, 150576, 183926, 222531, 266916, 317631, 375251, 440376, 513631, 595666, 687156, 788801, 901326, 1025481, 1162041, 1311806, 1475601, 1654276, 1848706, 2059791, 2288456, 2535651

Offset: 0

Author

N. J. A. Sloane, Jun 07 2015

Keywords

• nonn
• easy

Comments

If x is replaced by x^5, this is the Molien series for the Heisenberg group H(5).

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Yang-Hui He and John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Programs

Magma

I:=[1,6,41,156,426]; [n le 5 select I[n] else 5*Self(n-1)-10*Self(n-2)+10*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..45]]; // Vincenzo Librandi, Jun 08 2015

Mathematica

CoefficientList[Series[(1 +x +21x^2 +x^3 +x^4)/(1-x)^5, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 08 2015 *)
LinearRecurrence[{5,-10,10,-5,1},{1,6,41,156,426},40] (* Harvey P. Dale, Dec 01 2017 *)

Sage

[1 + 5*n*(n+1)*(5*n^2+5*n+2)/24 for n in (0..50)] # G. C. Greubel, Mar 24 2022

Formula

G.f.: (1 + x + 21*x^2 + x^3 + x^4)/(1-x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - Vincenzo Librandi, Jun 08 2015

a(n) = (25/24)*n^4 + (25/12)*n^3 + (35/24)*n^2 + (5/12)*n + 1 = 1 + 5*n*(n+1)*(5*n^2 + 5*n + 2)/24 = 1 + 5*A006322(n). - R. J. Mathar, Nov 09 2018

E.g.f.: (1/24)*(24 + 120*x + 360*x^2 + 200*x^3 + 25*x^4)*exp(x). - G. C. Greubel, Mar 24 2022